Theory

Unconstrained and constrained maxima and minima

Where to look for the extrema of a function of several variables, how to classify them with the Hessian matrix, how to handle compact sets and constraints with Lagrange multipliers.

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Unconstrained maxima and minima
Critical points and Fermat’s theorem

Unconstrained maxima and minima

Maximum and minimum, local and absolute

Let $f : A \to R$ with $A \subseteq R^{2}$ and let $(x_{0}, y_{0}) \in A$ .

$(x_{0}, y_{0})$ is an absolute maximum point if $f (x, y) \leq f (x_{0}, y_{0})$ for every $(x, y) \in A$ .
$(x_{0}, y_{0})$ is a local maximum point if there exists $r > 0$ such that $f (x, y) \leq f (x_{0}, y_{0})$ for every $(x, y) \in A$ with $(x - x_{0})^{2} + (y - y_{0})^{2} < r$ .

With the opposite inequalities one defines the absolute minimum and the local minimum; maxima and minima are called extrema. The definitions are the same as in the one-variable case, with the circular neighborhood in place of the interval, and they extend without changes to $R^{n}$ . An absolute extremum is also local; the converse is not true, therefore the two notions require different tools: differential calculus for local extrema, and a comparison of the values at the candidates for absolute ones.

Example

$f (x, y) = x^{2} + y^{2}$ has an absolute minimum point at $(0, 0)$ , since $f (x, y) \geq 0 = f (0, 0)$ for every $(x, y) \in R^{2}$ . The same function has no maximum, because along the $x$ -axis we have $f (x, 0) = x^{2} \to + \infty$ .

Critical points and Fermat’s theorem

1. Critical point

Let $f$ have partial derivatives at $(x_{0}, y_{0})$ , an interior point of the domain. The point $(x_{0}, y_{0})$ is critical (or stationary) for $f$ if the gradient vanishes, that is, if the two conditions are satisfied simultaneously:

{f_{x} (x_{0}, y_{0}) = 0 f_{y} (x_{0}, y_{0}) = 0

Finding the critical points amounts to solving this system of two equations in two unknowns. At critical points the tangent plane to the graph is horizontal.

2. Fermat's theorem in several variables

Let $(x_{0}, y_{0})$ be a local extremum point for $f$ , interior to the domain, and suppose that $f$ has partial derivatives at $(x_{0}, y_{0})$ . Then $(x_{0}, y_{0})$ is a critical point:

\nabla f (x_{0}, y_{0}) = (0, 0) .

The theorem says where to look for unconstrained extrema: among the critical points, in addition to any interior points where $f$ is not differentiable and to the points of the boundary of the domain. For $f (x, y) = x^{2} + y^{2} - 4 x$ the system $f_{x} = 2 x - 4 = 0$ , $f_{y} = 2 y = 0$ has the unique solution $(2, 0)$ , hence any unconstrained extremum is at $(2, 0)$ .

Proof

Let $(x_{0}, y_{0})$ be a local maximum point (for the minimum the argument is the same). The one-variable function $g (t) = f (t, y_{0})$ is defined in a neighborhood of $t = x_{0}$ , because the point is interior to the domain, and it is differentiable at $x_{0}$ with $g^{'} (x_{0}) = f_{x} (x_{0}, y_{0})$ . Since $f$ has a local maximum at $(x_{0}, y_{0})$ , the restriction $g$ has a local maximum at the interior point $x_{0}$ , hence by Fermat’s theorem in one variable $g^{'} (x_{0}) = 0$ , that is $f_{x} (x_{0}, y_{0}) = 0$ . Applying the same argument to $h (t) = f (x_{0}, t)$ we obtain $f_{y} (x_{0}, y_{0}) = 0$ .

3. Saddle point

A critical point that is not a local extremum point is called a saddle point. The name comes from the graph of $f (x, y) = x^{2} - y^{2}$ : the origin is critical, but $f (x, 0) = x^{2} > 0$ and $f (0, y) = - y^{2} < 0$ for $x, y \neq = 0$ , hence in every neighborhood of the origin the function takes values both greater and smaller than $f (0, 0) = 0$ and the origin is not an extremum. Fermat’s theorem provides a necessary condition, not a sufficient one, therefore after finding the critical points a tool is needed to classify them.

f (x, y) = x^{2} - y^{2}

is positive in the shaded region and negative elsewhere: in every neighborhood of the origin it takes both signs

The theory continues with 6 more sections (the hessian matrix and the classification of critical points, the degenerate case direct study of the sign, absolute extrema on compact sets…), plus all worked examples, the solutions and the PDF.

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